Mathematical models have been proposed for some neurological disorders characterized by a propagating process across the brain, of which one example is Cortical Spreading Depression (CSD). CSD is a massive but temporary perturbation in the cortical ionic homeostasis leading to a depression of neuronal activity that spreads through the cortex and the other gray matter regions in the brain. CSD has been suggested to be associated with migraines with aura, traumatic brain injury, epilepsy, ischemic stroke and subarachnoid hemorrhage. Another such disorder is epilepsy, in which many neurons may fire simultaneously, in a propagating wave originating at a focus. Medical imaging with different modalities (MRI, fMRI, dMRI, SPECT, PET, etc.) has been extremely helpful in understanding the effects in brain structure of neurological disorders. However, there has been less progress in understanding the functional characteristics of disorders where such propagating processes play a role, in a subject-specific manner. Doing this requires understanding how the propagating process moves across the surface of the cerebral cortex, and also taking into account the local and long-range connections between brain regions. Furthermore, there might also be additional measurements of other relevant aspects of function, such as external (EEG) or internal (ECoG) electrical activity, as the propagating process takes place.
Mathematically, the process of CSD has been modeled as a system of coupled reaction-diffusion equations for the different ions (K+, Ca2+, etc.), incorporating the physics of diffusion of ions in the extracellular space, along with the ionic dynamics controlled with additional gating variables. Computational models employing these equations have been used to explain the dynamics of the solitary wave, along with more complex features like the formation of spiral waves. However, very limited research has been done on real subject anatomies owing to multiple difficulties including lack of access to detailed cortical geometries; difficulty of modeling the dynamics over these complex geometric surfaces; and large computational demands of the numerical solver. Accordingly, it is desirable to produce a subject-specific model that provides accurate modeling of CSD (or other neurological disorders), while limiting the computational demand of the model's processing.